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ACTSC445 Study Guide - Quiz Guide: Likelihood-Ratio Test, Independent And Identically Distributed Random Variables, Random VariableExam


Department
Actuarial Science
Course Code
ACTSC445
Professor
Jiahua Chen
Study Guide
Quiz

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ACTSC 445/845 – Fall 2014
ASSIGNMENT 3 DUE:10:55 am Thursday November 27th
Hand in to Yunran Wei in the TA office.
Notes:
You may work on this assignment individually or in groups of up to three people. If
you work in a group, you should submit a single solution with each students’ name
and ID number.
If you work in a group, each student in the group will receive the same mark.
There will be no extra credit for students who work individually.
For use in both questions:
Download a data set of weekly returns from Yahoo.com for the period of 1990-2013 for two
stocks, A and B. Use financial related industries (banks, insurers, etc). See the note at the
end of the assignment for details of how to do this.
1. For stock A and stock B, separately:
We are interested in studying extreme outcomes of the negative logreturns, per
$100 invested, which is denoted by Xi=100 log(Si+1/Si) for the ith week, where
Siis your stock price. You may assume that the log-returns are iid with a common
distribution F.
Let Mj,j= 1,2, ..., n denote the maximum values of Xiin the jth block, where each
block is 26 weeks long.
(a) Generate a histogram of the observed Mj.
(b) Using maximum likelihood, estimate the GEV parameters for the distribution.
(c) Compare the data and the tted model with a Q-Q plot, and comment on the t.
(d) Use the likelihood ratio test to compare the simpler model, γ= 0 with the model
γ > 0, and comment on the results.
(e) Use sample mean excess plot to nd an appropriate threshold uso that the ex-
ceedances can be tted with a generalized Pareto distribution (GPD).
(f) Fit the GPD to the exceedances of the negative logreturns.
(g) Test the hypothesis γ= 0, using the likelihood ratio test.
(h) Based on the GPD you obtained in the previous steps, produce a mean excess
plot of the negative logreturns.
(i) Develop estimates for V aR0.99(X) and CTE0.99(X) by using the PGD you devel-
oped, where Xdenotes the negative logreturn random variable.
(j) Compare the results for Stock A and Stock B.
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